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Theorem ecovicom 6505
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)
Hypotheses
Ref Expression
ecovicom.1  |-  C  =  ( ( S  X.  S ) /.  .~  )
ecovicom.2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
ecovicom.3  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
ecovicom.4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  D  =  H )
ecovicom.5  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  G  =  J )
Assertion
Ref Expression
ecovicom  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
Distinct variable groups:    x, y, z, w, A    z, B, w    x,  .+ , y, z, w    x,  .~ , y, z, w    x, S, y, z, w    z, C, w
Allowed substitution hints:    B( x, y)    C( x, y)    D( x, y, z, w)    G( x, y, z, w)    H( x, y, z, w)    J( x, y, z, w)

Proof of Theorem ecovicom
StepHypRef Expression
1 ecovicom.1 . 2  |-  C  =  ( ( S  X.  S ) /.  .~  )
2 oveq1 5749 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( A  .+  [ <. z ,  w >. ]  .~  ) )
3 oveq2 5750 . . 3  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
) )
42, 3eqeq12d 2132 . 2  |-  ( [
<. x ,  y >. ]  .~  =  A  -> 
( ( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  <->  ( A  .+  [ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A ) ) )
5 oveq2 5750 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( A  .+  [ <. z ,  w >. ]  .~  )  =  ( A  .+  B ) )
6 oveq1 5749 . . 3  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( [ <. z ,  w >. ]  .~  .+  A )  =  ( B  .+  A ) )
75, 6eqeq12d 2132 . 2  |-  ( [
<. z ,  w >. ]  .~  =  B  -> 
( ( A  .+  [
<. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  A
)  <->  ( A  .+  B )  =  ( B  .+  A ) ) )
8 ecovicom.4 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  D  =  H )
9 ecovicom.5 . . . 4  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  G  =  J )
10 opeq12 3677 . . . . 5  |-  ( ( D  =  H  /\  G  =  J )  -> 
<. D ,  G >.  = 
<. H ,  J >. )
1110eceq1d 6433 . . . 4  |-  ( ( D  =  H  /\  G  =  J )  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
128, 9, 11syl2anc 408 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  ->  [ <. D ,  G >. ]  .~  =  [ <. H ,  J >. ]  .~  )
13 ecovicom.2 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  [ <. D ,  G >. ]  .~  )
14 ecovicom.3 . . . 4  |-  ( ( ( z  e.  S  /\  w  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1514ancoms 266 . . 3  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  )  =  [ <. H ,  J >. ]  .~  )
1612, 13, 153eqtr4d 2160 . 2  |-  ( ( ( x  e.  S  /\  y  e.  S
)  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( [ <. x ,  y >. ]  .~  .+ 
[ <. z ,  w >. ]  .~  )  =  ( [ <. z ,  w >. ]  .~  .+  [
<. x ,  y >. ]  .~  ) )
171, 4, 7, 162ecoptocl 6485 1  |-  ( ( A  e.  C  /\  B  e.  C )  ->  ( A  .+  B
)  =  ( B 
.+  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   <.cop 3500    X. cxp 4507  (class class class)co 5742   [cec 6395   /.cqs 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fv 5101  df-ov 5745  df-ec 6399  df-qs 6403
This theorem is referenced by:  addcomnqg  7157  mulcomnqg  7159  addcomsrg  7531  mulcomsrg  7533  axmulcom  7647
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